I think it is more straightforward to walk through the entire phrase without worrying about splitting on spaces. Instead keep track of whether or not you've just seen a space. You only want to add the character after just seeing a space.

def create_password(phrase):
q = "'" # quotations
dq = '"' # double quotes
#Initialize the password to be an empty string
password = ""
#We are at the start of a new word (want to add first index to password)
new_word = True
#Walk through every character in the phrase
for char in phrase:
#We only want to add char to password if the following is all true:
#(1) It's a letter or number
#(2) It's at the start of a new word
#(3) It's not a single quote
#(4) It's not a double quote
if char.isalnum and new_word:
if char != q and char != dq:
password += char
new_word = False #<-- After adding char, we are not at a new word
#If we see a space then we are going to be at a new word
elif char == " ":
new_word = True
return password
p = create_password('Stop and smell the 350 "roses"')
print(p)